# Combinatorial and integer optimization

**DOI:**https://doi.org/10.1007/1-4020-0611-X_129

## INTRODUCTION

Combinatorial optimization problems are concerned with the efficient allocation of limited resources to meet desired objectives when the values of some or all of the variables are restricted to be integral. Constraints on basic resources, such as labor, supplies, or capital restrict the possible alternatives that are considered feasible. Still, in most such problems, there are many possible alternatives to consider and one overall goal determines which of these alternatives is best. For example, most airlines need to determine crews chedules which minimize the total operating cost; automotive manufacturers may want to determine the design of a fleet of cars which will maximize their share of the market; a flexible manufacturing facility needs to schedule the production for a plant without having much advance notice as to what parts will need to be produced that day. In today's changing and competitive industrial environment the difference between using a quickly derived...

## References

- [1]Ahuja, R. K., Magnanti, T. L., and Orlin, J. (1993). Network Flows: Theory, Algorithms and Applications. Prentice-Hall, New Jersey.Google Scholar
- [2]Ahuja, R. K., Magnanti, T. L., Orlin, J. B., and Reddy, M. R. (1995). “Applications of Network Optimization,” Chapter 1 of Handbooks of Operations Research and Management Science, Vol. 7,
*Network Models*, M. O. Ball, T. L. Magnanti, C. L. Monma, and G. L. Nemhauser, Elsevier North Holland, 1–84. Google Scholar - [3]Anderson, E. D. and Anderson, K. D. (1995). “Presolving in Linear Programing,” Mathematical Programming, 71, 221–245.Google Scholar
- [4]Balas, E. and Padberg, M. (1976). “Set Partitioning: A Survey,” SIAM Review, 18, 710–760.Google Scholar
- [5]Barahona, M., Grötschel, M., Jünger, G., and Reinelt, G. (1988). “An Application of Combinatorial Optimization to Statistical Physics and Circuit Layout Design,” Operations Research, 36, 493–513.Google Scholar
- [6]Barnhart, C., Johnson, E. D., Nemhauser, G. L., Savelsbergh, M. W.P., and Vance, P. H. (1998). “Branch and Price Column Generation for Solving Huge Integer Programs,” Operations Research, 46, 316–329.Google Scholar
- [7]Benders, J. F. (1962). “Partitioning Procedures for Solving Mixed-variables Programming Problems,” Numerische Mathematik, 4, 238–252.Google Scholar
- [8]Beyer, D. and Ogier, R. (1991). “Tabu Learning: a Neural Network Search Method for Solving Nonconvex 0ptimization Problems,” Proceedings of the International Joint Conference on Neural Networks. IEEE and INNS, Singapore. Google Scholar
- [9]Bramel, J. and Simchi-Levi, D. (1998). On the effectiveness of set covering formulations for the vehicle routing problem with time windows” Technical Report, Dept. of Industrial Engineering and Mgmt. Sci., Northwestern University, Evanston, Illinois.Google Scholar
- [10]Brearley, A. L., Mitra, G., and Williams, H. P. (1975). “Analysis of mathematical programming problems prior to applying the simplex method,” Mathematical Programming, 8, 54–83.Google Scholar
- [11]Chan, L. M.A., Muriel, A., and Simchi-Levi, D. (1998). “Parallel machine scheduling, linear programming and parameter list scheduling heuristics,” Technical Report, Dept. of Industrial Engineering and Mgmt. Sci., Northwestern University, Evanston, Illinois.Google Scholar
- [12]Chang, L. M.A., Simchi-Levi, D., and Bramel, J. (1998). “Worst-case analysis, linear programming and the bin-packing problem,” Technical Report, Dept of Industrial Engineering and Mgmt. Sci., Northwestern University, Evanston, Illinois.Google Scholar
- [13]Chopra, S., Gorres, E., and Rao, M. R. (1992). “Solving the Steiner tree problem on a graph using branch and cut,” ORSA Journal on Computing, 4, 320–335.Google Scholar
- [14]Cooper, M. W. and Farhangian, K. (1985). “Multicriteria optimization for nonlinear integer-variable problems,” Large Scale Systems, 9, 73–78.Google Scholar
- [15]Cornuejols, G., Nemhauser, G. L., and Wolsey, L. A. (1980). “Worst case and probabilistic analysis of algorithms for a location problem,” Operations Research, 28, 847–858.Google Scholar
- [16]Desrochers, M. and Sosumis, F. (1989) “A column generation approach to the urban transit crew scheduling problem,” Transportation Science, 23, 1–13.Google Scholar
- [17]Fisher, M. L. (1981). “The Lagrangian method for solving integer programming problems,” Management Science, 27, 1–18.Google Scholar
- [18]Garey, M. R. and Johnson, D. S. (1979). Computers and Intractibility: A Guide to the Theory of NP-Completeness. W.H. Freeman, San Francisco.Google Scholar
- [19]Gilmore, P. C. and Gomory, R. E. (1961). “A linear programming approach to the cutting stock problem,” Operations Research, 9, 849–859.Google Scholar
- [20]Glover, F. and Laguna, M. (1993). “Tabu Search,” in Modern Heuristic Techniques for Combinatorial Optimization, C. R. Reeves, ed. Scientific Pubs., Oxford, England, 71–140.Google Scholar
- [21]Gomory, R. E. (1958). “Outline of an algorithm for integer solution to linear programs,” Bulletin American Mathematical Society, 64, 275–278.Google Scholar
- [22]Gomory, R. E. (1960). “Solving linear programming problems in integers,” Combinatorial Analysis, R. E. Bellman and M. Hall, Jr., eds., American Mathematical Society, 211–216. Google Scholar
- [23]Grötschel, M. (1992). “Discrete mathematics in manufacturing,” Preprint SC92-3, ZIB. Google Scholar
- [24]Grötschel, M., Lovasz, L., and Schrijver, A. (1988). Geometric Algorithms and Combinatorial Optimization, Springer, Berlin.Google Scholar
- [25]Grötschel, M., Monma, C. L., and Stoer, M. (1989). “Computational results with a cutting plane algorithm for designing communication networks with low-connectivity constraint,” Report No. 187, Schwerpunktprogramm der Deutschen Forschungs-gemeinschaft, Universität Augsburg. Google Scholar
- [26]Guignard, M. and Spielberg, K. (1981). “Logical Reduction Methods in Zero-one Programming: Minimal Preferred Inequalities,” Operations Research, 29, 49–74.Google Scholar
- [27]Guignard, M. and Kim, S. (1987). “Lagrangian decomposition: a model yielding stronger Lagrangian bounds,” Mathematical Programming, 39, 215–228.Google Scholar
- [28]Hansen, P. (1986). “The steepest ascent mildest descent heuristic for combinatorial programming,” Proceedings of Congress on Numerical Methods in Combinatorial Optimization, Capri, Italy. Google Scholar
- [29]Hansen, P., Jaumard, B., and Mathod, V. (1993) “Constrained nonlinear 0–1 programming” INFORMS Journal on Computing, 5, 97–119.Google Scholar
- [30]Hoffman, K. L. and Padberg, M. (1985). “LP-based Combinatorial Problem Solving,” Annals Operations Research, 4, 145–194.Google Scholar
- [31]Hoffman, K. L. and Padberg, M. (1991). “Improving the LP-representation of zero-one linear programs for branch-and-cut,” ORSA Journal Computing, 3, 121–134.Google Scholar
- [32]Hoffman, K. L. and Padberg, M. (1993). “Solving air-line crew scheduling problems by branch-and-cut,” Management Science, 39, 657–682.Google Scholar
- [33]Johnson, E. L. and Powell, S. (1978). “Integer programming codes,” Design and Implementation of Optimization Software (ed. H. J. Greenberg), NATO Advanced Study Institute Series, Sijthoff & Noordhoff, 225–248. Google Scholar
- [34]Karp, R. M. (1976). “Probabilistic analysis of partitioning algorithms for the traveling salesman problem,” in Algorithms and Complexity: New Directions and Recent Results (J. F. Traub, ed.) Academic Press, New York, 1–19.Google Scholar
- [35]Land, A. H. and Doig, A. G. (1960). “An automatic method for solving discrete programming problems,” Econometrica, 28, 497–520.Google Scholar
- [36]Magnanti, T. L. and Vachani, R. (1990). “A strong cutting plane algorithm for production scheduling with changeover costs,” Operations Research, 38, 456–473.Google Scholar
- [37]Martello, S. and Toth, P. (1990). Knapsack Problems, John Wiley, New York.Google Scholar
- [38]Markowitz, H. and Manne, A. (1957). “On the solution of discrete programming problems,” Econometrica, 2, 84–110.Google Scholar
- [39]McAloon, K. and Tretkoff, C. (1996). Optimization and Computational Logic, John Wiley, New York.Google Scholar
- [40]Mühlenbein, H. (1992). “Parallel genetic algorithms in combinatorial optimization,” Computer Science and Operations Research, Osman Blaci, ed., Pergamon Press, New York.Google Scholar
- [41]Nemhauser, G. L. and Wolsey, L. A. (1988). Integer and Combinatorial Optimization, John Wiley, New York.Google Scholar
- [42]Padberg, M. (1979). “Covering, packing and knapsack problems,” Mathematical Programming, 47, 19–46. Google Scholar
- [43]Padberg, M. (1995). Linear Optimization and Extensions, Springer Verlag, Heidelberg.Google Scholar
- [44]Padberg, M. and Rijal, M. P. (1996) Location Scheduling and Design in Integer Programming,” Kluwer Academic, Norwell, Massachusetts.Google Scholar
- [45]Padberg, M. and Rinaldi, G. (1991). “A branch–and cut algorithm for the resolution of large-scale symmetric traveling salesman problems,” SIAM Review, 33, 60–100.Google Scholar
- [46]Parker, R. G. and Rardin, R. L. (1988). Discrete Optimization, Academic Press, San Diego.Google Scholar
- [47]Pochet, Y. and Wolsey, L. A. (1991). “Solving Multi-item Lot Sizing Problems Using Strong Cutting Planes,” Management Science, 37, 53–67.Google Scholar
- [48]Rinooy Kan, A. H.G. (1986). “An introduction to the analysis of approximation algorithms,” Discrete Applied Mathematics, 14, 111–134.Google Scholar
- [49]Savelsbergh, M. W.P. (1997). “A branch-and-price algorithm for the generalized assignment problem,” Operations Research, 45, 831–841.Google Scholar
- [50]Schrijver, A. (1984). Linear and Integer Programming, John Wiley, New York.Google Scholar
- [51]Van Roy, T. J. and Wolsey, L. A. (1987). “Solving Mixed Integer Programming Problems Using Automatic Reformulation,” Operations Research, 35, 45–57.Google Scholar
- [52]Weingartner, H. (1963). Mathematical Programming and the Analysis of Capital Budgeting Problems, Prentice Hall, Englewood Cliffs, New Jersey.Google Scholar
- [53]Weyl, H. (1935). “Elementare Theorie der konvexen Polyheder,” Comm. Math. Helv, 7, 290 (Translated in
*Contributions to the Theory of Games*, 1(3), 1950).Google Scholar - [54]Williams, H. P. (1985). Model Building in Mathematical Programming, 2nd ed. John Wiley, New York.Google Scholar